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Spherical fast multiscale approximation by locally compact orthogonal wavelets

  • Frank Bauer
  • Martin Gutting
Original Paper

Abstract

Using a stereographical projection to the plane we construct an \({\mathcal{O}(N\log(N))}\) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere or a local subset of it. In fact, the sphere is not treated all at once, but is split into subdomains whose results are combined afterwards. After choosing the center of the area of interest the scattered data points are mapped from the sphere to the tangential plane through that point. By combining a k-nearest neighbor search algorithm and the two dimensional fast wavelet transform a fast approximation of the data is computed and mapped back to the sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 min on a standard computer using our MATLAB® implementation. The method is very flexible and allows the application of the full range of two dimensional wavelets.

Keywords

Orthogonal wavelets on the sphere Spherical multiresolution analysis Fast wavelet transform Scattered data Approximation of spherical functions 

Mathematics Subject Classification (2000)

42C40 43A90 65D10 86-08 65T60 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Fuzzy Logic Laboratorium Linz-HagenbergUniversity of LinzHagenbergAustria
  2. 2.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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